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A cylinder over a torus is by definition $S^1 \times S^1 \times I$ , here $I=[0,1]$.

One way to visualize it is to thick a torus in $\mathbb{R}^3$. ( $S^1 \times I$ is an annulus, and revolve it (the first $S^1$).) The other way is the compliment of a neighborhood of Hopf links in $S^3$. Suppose that the neiborhood lies in $\mathbb{R}^3 \subset \mathbb{R}^3 \cup {\infty}=S^3$. The second one is not actually a visualization since we (at least I) can't see the point at infinity.

We provide them with orientation induced by right hand orientation in $\mathbb{R}^3$ and $S^3$ respectively.

Question; So there is a orientation preserving homeomorphism between them but how does it look like? How can I visualize the transition from one to another? Especially I want to see how boundaries are transformed.

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  • $\begingroup$ The complement of the Hopf link in $S^3$ is the same thing as the complement of the union of unit circle in the $x-y$ plane and the $z$-axis in ${\bf R}^3$. $\endgroup$
    – Paul
    Feb 17, 2012 at 14:08
  • $\begingroup$ The stereographic projection $(x,y,z)\mapsto (2\cdot x,2\cdot y, 2\cdot z,x^2+y^2+z^2-1)/(1+x^2+y^2+z^2)$ gives an explicit map $\mathbb R^3\to\mathbb S^3$. You could see what is the image of your thick torus. The coordinate lines of the natural parametrization become circle arcs... Have fun. $\endgroup$ Feb 17, 2012 at 14:37

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